Let's analyze the provided linear function and arithmetic sequence step by step.
Linear Function Analysis
The linear function data points are:
x y
-1 10
0 14
1 18
2 22
Rate of Change: To find the rate of change (slope) of the linear function, we can calculate the change in \( y \) divided by the change in \( x \):
- From \( (0, 14) \) to \( (1, 18) \): \[ \text{Rate of Change} = \frac{18 - 14}{1 - 0} = \frac{4}{1} = 4 \] Thus, the rate of change of the linear function is 4.
Y-Intercept: The y-intercept is the value of \( y \) when \( x = 0 \). From the data, we see:
- When \( x = 0 \), \( y = 14 \). Thus, the y-intercept of the linear function is 14.
Arithmetic Sequence Analysis
The explicit formula for the arithmetic sequence is given as:
\[ a_n = 10 + 4(n-1) \]
Rate of Change (Common Difference): The common difference for this sequence can be derived from the formula. The coefficient of \( (n-1) \) is \( 4 \), which indicates that each term increases by \( 4 \). Thus, the rate of change (common difference) of the arithmetic sequence is 4.
First Value of the Arithmetic Sequence: To find the first value, set \( n = 1 \): \[ a_1 = 10 + 4(1-1) = 10 + 0 = 10 \] Thus, the first value of the arithmetic sequence is 10.
Summary Comparison
- Rate of Change of the Linear Function: 4
- Y-Intercept of the Linear Function: 14
- Rate of Change (Common Difference) of the Arithmetic Sequence: 4
- First Value of the Arithmetic Sequence: 10
Conclusion: The rate of change of the linear function (4) is equal to the rate of change of the arithmetic sequence (4). Arithmetic sequences can indeed be represented as linear functions, since both display a constant rate of change.
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